Finite Group Databasebeta

← #13177545

Subgroups — #13177545

Order 1728.

2596 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×#6292C211yesmaximal, characteristic, complemented
3#11723978C211yesmaximal, characteristic, complemented
4C32×#935C211yesmaximal, characteristic, complemented
5#11723639C211yesmaximal, characteristic, complemented
6#11723636C212yesmaximal, complemented
7#11723624C212yesmaximal, complemented
8C3⋊D3×#217C211yesmaximal, characteristic, complemented
9#11723605C211yesmaximal, characteristic, complemented
10#11723482C212yesmaximal, complemented
11#11723417C212yesmaximal, complemented
12#10596124—31nomaximal
13#10595234—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
15C23×C3×C3⋊D3C2211yescharacteristic
16C23×C3×C3⋊D3C2211yescharacteristic, complemented
17C2×C33×D4C2211yescharacteristic
18C2×#6312C2212yescomplemented
19C2×C3×#638C2211yescharacteristic
20C2×C3×#638C2211yescharacteristic, complemented
21C3×D4×C3⋊D3—24no—
22C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
23C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
24C22×C32×Dic3C2211yescharacteristic, complemented
25C2×#6292—24no—
26C2×#6292C2212yes—
27C2×#6292C2212yescomplemented
28C2×#6292C2212yescomplemented
29C2×Dic3×C3⋊D3C2212yescomplemented
30#88864C2211yescharacteristic, complemented
31#88859C2212yescomplemented
32C32×#217C2211yescharacteristic, complemented
33C32×#217C2212yescomplemented
34C32×Dic3⋊C4C2212yescomplemented
35#88806C2212yescomplemented
36#88804C2212yescomplemented
37#88803C2212yescomplemented
38#88803C2212yescomplemented
39C2×D4×C3⋊D3D311yescharacteristic, complemented
40C22×C4⋊(C3⋊D3)—31no—
41C2×C3×D3×D4—34no—
42C22×#626—34no—
43C32×#123—31no—
44#64146—31no—
45#64146—34no—
46#64146—64no—
47#64138—31no—
48#64104—32no—
49#64103—32no—
50#64093—32no—
51#64091—31no—
52#64091—32no—
53C22⋊C4×C3⋊D3—31no—
54C3×#935D314yescomplemented
55#63981—34no—
56#63977—38no—
57#63971—38no—
58D3×#217—34no—
59#63961—34no—
60#63886—38no—
61#63850—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C3⋊(C3⋊D3)—22no—
65C22×C3⋊(C3⋊D3)—42no—
66C22×C3⋊(C3⋊D3)C2312yes—
67C22×C3×C3⋊D3—22no—
68C22×C3×C3⋊D3—42no—
69C22×C3×C3⋊D3—42no—
70C22×C3×C3⋊D3D414yescomplemented
71C22×C3×C3⋊D3C2311yescharacteristic
72C22×C3×C3⋊D3C2311yescharacteristic
73C22×C3×C3⋊D3C2311yescharacteristic
74C33×D4—22no—
75C2×C33×C4C2311yescharacteristic
76#6312—42no—
77C2×C3⋊(C3⋊Dic3)C2312yes—
78C3×#638—22no—
79C3×#638—22no—
80C2×C3×C3⋊Dic3C2311yescharacteristic
81C3×C4⋊(C3⋊D3)—22no—
82C3×C4×C3⋊D3—22no—
83C2×C32×Dic3—22no—
84C2×C32×Dic3C2312yes—
85C2×C32×Dic3C2312yes—
86#6292—24no—
87#6292—24no—
88#6292—42no—
89Dic3×C3⋊D3—42no—
90#5804—91no—
91#5653—94no—
92C23×C3⋊D3—31no—
93C23×C3⋊D3—34no—
94C23×C3⋊D3—64no—
95C23×C3⋊D3C2×D311yescharacteristic
96C23×C3⋊D3C2×D311yescharacteristic, complemented
97C23×C3×D3—34no—
98C23×C3×D3—34no—
99C2×C32×D4—24no—
100C2×C32×D4C2×D314yes—
101C2×C32×D4C2×D311yescharacteristic
102C22×C32×C4—31no—
103C2×#638—32no—
104C2×#638—38no—
105C2×#638—68no—
106C2×#638C2×D311yescharacteristic
107C2×#638C2×D311yescharacteristic, complemented
108D4×C3⋊D3—24no—
109C2×C4⋊(C3⋊D3)—32no—
110C2×C4⋊(C3⋊D3)—32no—
111C2×C4⋊(C3⋊D3)—32no—
112C2×C4⋊(C3⋊D3)—64no—
113C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
114C2×C4×C3⋊D3—32no—
115C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
116C2×C3×#67—34no—
117C2×C3×#67—34no—
118C22×C3×Dic3C2×D314yescomplemented
119C3×D3×D4—616no—
120C2×C3×D12—34no—
121C2×C3×C4×D3—34no—
122C2×#626—38no—
123C2×#626—38no—
124C2×#626—38no—
125C2×#626—616no—
126C2×D3×Dic3—38no—
127C32×C4⋊C4—32no—
128C32×C22⋊C4—31no—
129C32×C22⋊C4—32no—
130#3757—31no—
131#3757—34no—
132#3757—64no—
133#3752—32no—
134#3752—32no—
135#3752—32no—
136#3752—38no—
137#3752—32no—
138#3752—68no—
139#3750—32no—
140C3×#217C2×D314yescomplemented
141C3×#217C2×D318yescomplemented
142C3×Dic3⋊C4C2×D318yescomplemented
143#3724—38no—
144#3722—38no—
145(C2×D3)⋊Dic3—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33—22no—
148C22×C33—21no—
149C22×C33—22no—
150C22×C33C2×D412yes—
151C22×C33C2411yescharacteristic
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3⋊(C3⋊D3)—22no—
154C2×C3⋊(C3⋊D3)—22no—
155C2×C3⋊(C3⋊D3)—44no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—21no—
159C2×C3×C3⋊D3—22no—
160C2×C3×C3⋊D3—44no—
161C2×C3×C3⋊D3—44no—
162C2×C3×C3⋊D3C2×D412yes—
163C2×C3×C3⋊D3C2×D412yes—
164C33×C4—21no—
165C3⋊(C3⋊Dic3)—22no—
166C3×C3⋊Dic3—21no—
167C32×Dic3—22no—
168C32×Dic3—22no—
169C22×C3×D4—91no—
170C22×#67—91no—
171C2×D3×D4—34no—
172C22×D12—94no—
173C3×#123—34no—
174#953—92no—
175C2×#217—91no—
176#939—92no—
177#939—92no—
178#937—94no—
179#937—91no—
180#937—91no—
181#937—184no—
182#935—91no—
183#935C3⋊D311yescharacteristic, complemented
184#929—94no—
185#929—92no—
186#895—98no—
187#894—98no—
188#884—98no—
189#882—94no—
190#882—98no—
191D3×C22⋊C4—94no—
192C23×C32—24no—
193C23×C32—24no—
194C23×C32C22×D314yes—
195C23×C32C22×D314yes—
196C23×C32C22×D311yescharacteristic
197C23×C32C22×D311yescharacteristic
198C22×C3⋊D3—22no—
199C22×C3⋊D3—32no—
200C22×C3⋊D3—38no—
201C22×C3⋊D3—42no—
202C22×C3⋊D3—42no—
203C22×C3⋊D3—68no—
204C22×C3⋊D3—62no—
205C22×C3⋊D3—68no—
206C22×C3⋊D3—128no—
207C22×C3⋊D3—122no—
208C22×C3⋊D3—128no—
209C22×C3⋊D3—1216no—
210C22×C3⋊D3#6714yescomplemented
211C22×C3⋊D3C22×D311yescharacteristic
212C22×C3⋊D3C22×D311yescharacteristic
213C22×C3⋊D3C22×D311yescharacteristic
214C22×C3×D3—34no—
215C22×C3×D3—34no—
216C22×C3×D3—316no—
217C22×C3×D3—34no—
218C22×C3×D3—68no—
219C22×C3×D3—128no—
220C22×C3×D3—128no—
221C32×D4—22no—
222C32×D4—28no—
223C32×D4—48no—
224C2×C32×C4—24no—
225C2×C32×C4—32no—
226C2×C32×C4—32no—
227C2×C32×C4—62no—
228C2×C32×C4C22×D314yes—
229C2×C32×C4C22×D311yescharacteristic
230#638—22no—
231#638—22no—
232#638—122no—
233#638—128no—
234#638—1216no—
235C2×C3⋊Dic3—32no—
236C2×C3⋊Dic3—38no—
237C2×C3⋊Dic3—68no—
238C2×C3⋊Dic3C22×D311yescharacteristic
239C4⋊(C3⋊D3)—22no—
240C4⋊(C3⋊D3)—64no—
241C4⋊(C3⋊D3)—64no—
242C4⋊(C3⋊D3)—122no—
243C4×C3⋊D3—22no—
244C4×C3⋊D3—122no—
245C3×#67—68no—
246C3×#67—68no—
247C2×C3×Dic3—28no—
248C2×C3×Dic3—34no—
249C2×C3×Dic3C22×D318yes—
250C2×C3×Dic3C22×D318yes—
251C3×D12—68no—
252C3×C4×D3—68no—
253#626—616no—
254#626—616no—
255#626—128no—
256D3×Dic3—128no—
257#534—271noSylow
258C2×C33—22no—
259C2×C33—41no—
260C2×C33C22×D411yescharacteristic
261C2×C33H(5,𝔽2)12yes—
262C3⋊(C3⋊D3)—42no—
263C3×C3⋊D3—22no—
264C3×C3⋊D3—22no—
265C3×C3⋊D3—41no—
266C24×C3—91no—
267C24×C3—91no—
268C23×D3—34no—
269C23×D3—34no—
270C23×D3—94no—
271C23×D3—91no—
272C23×D3—184no—
273C2×C3×D4—24no—
274C2×C3×D4—91no—
275C2×C3×D4—91no—
276C2×C3×D4—91no—
277C2×C3×D4—184no—
278C2×C3×D4D3214yes—
279C2×C3×D4C2×C3⋊D311yescharacteristic
280C22×C3×C4—34no—
281C22×C3×C4—91no—
282C2×#67—34no—
283C2×#67—34no—
284C2×#67—98no—
285C2×#67—92no—
286C2×#67—92no—
287C2×#67—92no—
288C2×#67—92no—
289C2×#67—188no—
290C2×#67—184no—
291C22×Dic3—92no—
292C22×Dic3C2×C3⋊D311yescharacteristic, complemented
293D3×D4—616no—
294C2×D12—34no—
295C2×D12—98no—
296C2×D12—98no—
297C2×D12—98no—
298C2×D12—1816no—
299C2×C4×D3—34no—
300C2×C4×D3—98no—
301C3×C4⋊C4—38no—
302C3×C22⋊C4—34no—
303C3×C22⋊C4—38no—
304#217—94no—
305#217—91no—
306#217—92no—
307#217—92no—
308#217—184no—
309#217C2×C3⋊D311yescharacteristic, complemented
310#217C2×C3⋊D312yescomplemented
311(C2×D3)⋊C4—98no—
312(C2×D3)⋊C4—98no—
313(C2×D3)⋊C4—98no—
314(C2×D3)⋊C4—98no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—92no—
317(C2×D3)⋊C4—188no—
318Dic3⋊C4—98no—
319Dic3⋊C4—92no—
320Dic3⋊C4C2×C3⋊D312yescomplemented
321C22×C32—22no—
322C22×C32—22no—
323C22×C32—21no—
324C22×C32—24no—
325C22×C32—28no—
326C22×C32—28no—
327C22×C32—24no—
328C22×C32—28no—
329C22×C32—48no—
330C22×C32—48no—
331C22×C32—44no—
332C22×C32D3×D418yes—
333C22×C32C2×#6712yes—
334C22×C32C23×D314yes—
335C22×C32C23×D311yescharacteristic
336C2×C3⋊D3—22no—
337C2×C3⋊D3—22no—
338C2×C3⋊D3—22no—
339C2×C3⋊D3—21no—
340C2×C3⋊D3—44no—
341C2×C3⋊D3—44no—
342C2×C3⋊D3—616no—
343C2×C3⋊D3—62no—
344C2×C3⋊D3—62no—
345C2×C3⋊D3—68no—
346C2×C3⋊D3—62no—
347C2×C3⋊D3—68no—
348C2×C3⋊D3—68no—
349C2×C3⋊D3—128no—
350C2×C3⋊D3—128no—
351C2×C3⋊D3—124no—
352C2×C3⋊D3—1216no—
353C2×C3⋊D3—1232no—
354C2×C3⋊D3C2×#6712yes—
355C2×C3⋊D3C2×#6712yes—
356C2×C3×D3—38no—
357C2×C3×D3—38no—
358C2×C3×D3—68no—
359C2×C3×D3—68no—
360C2×C3×D3—64no—
361C2×C3×D3—68no—
362C2×C3×D3—1216no—
363C2×C3×D3—1216no—
364C32×C4—21no—
365C32×C4—24no—
366C32×C4—44no—
367C32×C4—62no—
368C32×C4—62no—
369C3⋊Dic3—21no—
370C3⋊Dic3—62no—
371C3⋊Dic3—68no—
372C3⋊Dic3—128no—
373C3×Dic3—28no—
374C3×Dic3—28no—
375C3×Dic3—64no—
376C22×D4—91no—
377C22×D4—271no—
378#123—31no—
379#123—272no—
380#123—271no—
381#121—272no—
382#121—272no—
383#120—271no—
384#120—272no—
385#120—271no—
386C2×C22⋊C4—271no—
387C33#53411yescharacteristic, complemented, Sylow
388C23×C3—24no—
389C23×C3—24no—
390C23×C3—91no—
391C23×C3—91no—
392C23×C3—94no—
393C23×C3—91no—
394C23×C3—182no—
395C23×C3—362no—
396C23×C3—362no—
397C23×C3C2×D3214yes—
398C23×C3C2×D3214yes—
399C23×C3C22×C3⋊D311yescharacteristic
400C23×C3C22×C3⋊D311yescharacteristic
401C22×D3—34no—
402C22×D3—34no—
403C22×D3—316no—
404C22×D3—34no—
405C22×D3—68no—
406C22×D3—98no—
407C22×D3—92no—
408C22×D3—128no—
409C22×D3—128no—
410C22×D3—188no—
411C22×D3—188no—
412C22×D3—182no—
413C22×D3—368no—
414C22×D3—368no—
415C22×D3—362no—
416C22×D3—3616no—
417C3×D4—28no—
418C3×D4—22no—
419C3×D4—48no—
420C3×D4—182no—
421C3×D4—182no—
422C3×D4—182no—
423C2×C3×C4—24no—
424C2×C3×C4—38no—
425C2×C3×C4—38no—
426C2×C3×C4—68no—
427C2×C3×C4—91no—
428C2×C3×C4—182no—
429C2×C3×C4C2×D3214yes—
430C2×C3×C4C22×C3⋊D311yescharacteristic
431#67—68no—
432#67—68no—
433#67—184no—
434#67—184no—
435#67—362no—
436#67—368no—
437#67—362no—
438#67—3616no—
439C2×Dic3—22no—
440C2×Dic3—34no—
441C2×Dic3—98no—
442C2×Dic3—92no—
443C2×Dic3—188no—
444C2×Dic3—362no—
445C2×Dic3C22×C3⋊D312yes—
446C2×Dic3C22×C3⋊D312yes—
447D12—68no—
448D12—1816no—
449D12—1816no—
450D12—368no—
451C4×D3—68no—
452C4×D3—368no—
453C2×C32—22no—
454C2×C32—28no—
455C2×C32—24no—
456C2×C32—28no—
457C2×C32—48no—
458C2×C32—41no—
459C2×C32—44no—
460C2×C32—84no—
461C2×C32C2×D3×D414yes—
462C2×C32#100412yes—
463C2×C32#100918yes—
464C2×C32C22×#6711yescharacteristic
465C3⋊D3—22no—
466C3⋊D3—22no—
467C3⋊D3—41no—
468C3⋊D3—1216no—
469C3⋊D3—128no—
470C3⋊D3—122no—
471C3×D3—68no—
472C3×D3—68no—
473C3×D3—124no—
474C24—91no—
475C24—91no—
476C24—271no—
477C2×D4—91no—
478C2×D4—91no—
479C2×D4—91no—
480C2×D4—184no—
481C2×D4—272no—
482C2×D4—272no—
483C2×D4—272no—
484C2×D4—272no—
485C2×D4—544no—
486C2×D4D3×C3⋊D311yescharacteristic
487C22×C4—31no—
488C22×C4—91no—
489C22×C4—272no—
490C4⋊C4—32no—
491C4⋊C4—272no—
492C22⋊C4—31no—
493C22⋊C4—32no—
494C22⋊C4—272no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—271no—
498C22⋊C4—272no—
499C22×C3—28no—
500C22×C3—22no—
501C22×C3—28no—
502C22×C3—24no—
503C22×C3—21no—
504C22×C3—22no—
505C22×C3—24no—
506C22×C3—28no—
507C22×C3—48no—
508C22×C3—48no—
509C22×C3—44no—
510C22×C3—92no—
511C22×C3—92no—
512C22×C3—182no—
513C22×C3—182no—
514C22×C3—181no—
515C22×C3—182no—
516C22×C3—364no—
517C22×C3—364no—
518C22×C3D3×#6718yes—
519C22×C3D4×C3⋊D312yes—
520C22×C3C22×D3214yes—
521C22×C3C23×C3⋊D311yescharacteristic
522C2×D3—38no—
523C2×D3—38no—
524C2×D3—68no—
525C2×D3—68no—
526C2×D3—68no—
527C2×D3—64no—
528C2×D3—1216no—
529C2×D3—1216no—
530C2×D3—1816no—
531C2×D3—188no—
532C2×D3—188no—
533C2×D3—188no—
534C2×D3—182no—
535C2×D3—182no—
536C2×D3—182no—
537C2×D3—368no—
538C2×D3—368no—
539C2×D3—364no—
540C2×D3—3616no—
541C2×D3—3632no—
542C3×C4—24no—
543C3×C4—21no—
544C3×C4—44no—
545C3×C4—68no—
546C3×C4—68no—
547C3×C4—181no—
548Dic3—22no—
549Dic3—22no—
550Dic3—64no—
551Dic3—188no—
552Dic3—182no—
553Dic3—368no—
554C32—24no—
555C32#565314yescomplemented
556C32#580411yescharacteristic, complemented
557C23—91no—
558C23—91no—
559C23—94no—
560C23—91no—
561C23—182no—
562C23—272no—
563C23—362no—
564C23—362no—
565C23—542no—
566C23—1082no—
567C23C2×D3×C3⋊D311yescharacteristic
568C23C2×D3×C3⋊D311yescharacteristic
569D4—22no—
570D4—182no—
571D4—182no—
572D4—182no—
573D4—544no—
574D4—544no—
575D4—1082no—
576D4—1082no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—62no—
580C2×C4—91no—
581C2×C4—182no—
582C2×C4—272no—
583C2×C4—1082no—
584C2×C4C2×D3×C3⋊D311yescharacteristic
585C2×C3—22no—
586C2×C3—28no—
587C2×C3—28no—
588C2×C3—24no—
589C2×C3—48no—
590C2×C3—41no—
591C2×C3—44no—
592C2×C3—84no—
593C2×C3—182no—
594C2×C3—182no—
595C2×C3—361no—
596C2×C3#6431418yes—
597C2×C3C2×D3×#6714yes—
598C2×C3C2×D4×C3⋊D311yescharacteristic
599C2×C3#6436612yes—
600D3—68no—
601D3—68no—
602D3—124no—
603D3—3616no—
604D3—362no—
605D3—368no—
606C22—22no—
607C22—22no—
608C22—21no—
609C22—92no—
610C22—92no—
611C22—182no—
612C22—182no—
613C22—182no—
614C22—181no—
615C22—364no—
616C22—364no—
617C22—542no—
618C22—542no—
619C22—542no—
620C22—1084no—
621C22C3⋊D3×#6712yes—
622C22C22×D3×C3⋊D311yescharacteristic
623C4—21no—
624C4—62no—
625C4—62no—
626C4—181no—
627C4—542no—
628C3—24no—
629C3#1059523414yescomplemented
630C3#1059612411yescharacteristic, complemented
631C2—22no—
632C2—41no—
633C2—182no—
634C2—182no—
635C2—361no—
636C2—1082no—
637C2#1172458912yes—
638C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C22×#6292C2semidirect
3. #11723978C2semidirect
4. C32×#935C2semidirect
5. #11723639C2semidirect
6. #11723636C2semidirect
7. #11723624C2semidirect
8. C3⋊D3×#217C2semidirect
9. #11723605C2semidirect
10. #11723482C2semidirect
11. #11723417C2semidirect
14. C23×C3⋊(C3⋊D3)C22semidirect
16. C23×C3×C3⋊D3C22semidirect
18. C2×#6312C22semidirect
20. C2×C3×#638C22semidirect
22. C2×C3×C4⋊(C3⋊D3)C22semidirect
23. C2×C3×C4×C3⋊D3C22semidirect
24. C22×C32×Dic3C22semidirect
27. C2×#6292C22semidirect
28. C2×#6292C22semidirect
29. C2×Dic3×C3⋊D3C22semidirect
30. #88864C22semidirect
31. #88859C22semidirect
32. C32×#217C22semidirect
33. C32×#217C22semidirect
34. C32×Dic3⋊C4C22semidirect
35. #88806C22semidirect
36. #88804C22semidirect
37. #88803C22semidirect
38. #88803C22semidirect
39. C2×D4×C3⋊D3D3semidirect
54. C3×#935D3semidirect
70. C22×C3×C3⋊D3D4semidirect
96. C23×C3⋊D3C2×D3semidirect
107. C2×#638C2×D3semidirect
113. C2×C4⋊(C3⋊D3)C2×D3semidirect
115. C2×C4×C3⋊D3C2×D3semidirect
118. C22×C3×Dic3C2×D3semidirect
140. C3×#217C2×D3semidirect
141. C3×#217C2×D3semidirect
142. C3×Dic3⋊C4C2×D3semidirect
183. #935C3⋊D3semidirect
210. C22×C3⋊D3#67semidirect
292. C22×Dic3C2×C3⋊D3semidirect
309. #217C2×C3⋊D3semidirect
310. #217C2×C3⋊D3semidirect
320. Dic3⋊C4C2×C3⋊D3semidirect
387. C33#534semidirect
555. C32#5653semidirect
556. C32#5804semidirect
629. C3#10595234semidirect
630. C3#10596124semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×D4×C3⋊D32
C22×#62922
#117239782
C32×#9352
#117236392
#117236362
#117236242
C3⋊D3×#2172
#117236052
#117234822
#117234172
#105961243
#105952343